Abstract:
Inverse problems are ubiquitous in the engineering domain and often rely on computationally expensive forward models. For applications with societal or economical impact it is of major importance to quantify the uncertainties associated with the simulation results. A Bayesian formulation can inherently account for this uncertainty. However, for statistical inferences we need to repeatedly evaluate the posterior and thus a potentially costly forward model. Clearly this is computationally infeasible. A common practice is then to replace the expensive model by a cheaper surrogate. However, this introduces additional uncertainties on the posterior, since the model outputs are no longer accurate. To this end we propose a Bayesian multi-fidelity approach for inverse problems that can account for this additional uncertainty. The approach is based on carefully selected evaluations of the expensive high-fidelity model to construct a suitable predictor. This is done by finding a correlation between the high-fidelity and corresponding low-fidelity model outputs. The resulting predictive distribution then encapsulates the epistemic uncertainty introduced by the limited high- fidelity model evaluations. We shall further reformulate the Bayesian solution of the inverse problem when the computationally demanding forward model is re- placed by a surrogate. A numerical demonstration is then given based on the the Poisson equation with variable heat conductivity. We investigated the validity of the approach for different levels of fidelities and number of expensive model evaluations. It was found that the proposed framework could mostly provide reasonable confidence intervals of the solution of the inverse problem and thus capture the epistemic uncertainty induced by the limited number of high-fidelity solver evaluations.